On Random Fixed Points of Uniformly Lipschitza in Random Operators

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On Random Fixed Points of Uniformly Lipschitza in Random Operators

Yusra Jarallah Ajeel

Department of Mathematics, College of Science, Salwa Salman Abed

Department of Mathematics, College of Pure Science, Ibn Al-Haithem , Baghdad University

Saheb K. Alsaidy

Department of Mathematics, College of Science, AL-Mustansirya University

Abstract:

The purpose of this paper is to prove two random fixed point theorems for uniformly 1-lipschitzain random operator on non-star-shaped domain in Banach spaces. Our results is modified to results in [35].

Keywords: random operators, random fixed points, asymptotically non-expansive random operator, uniformly Lipschitzain random operator, Banach spaces.

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1. Introduction

The random approximations and random fixed point theorems are stochastic generalizations of usual approximations and fixed points theorems. Recently,many researchers are interested in this subject such as Beg and Shahzad [5], [6] who proved some theorems about the best approximations as applications for some stochastic fixed point theorems. And, Khan ,Thaheem and Hussain [14] are proved random fixed point theorems for nonexpansive random operators defined on a class of nonexpansive sets containing the subclass of starshaped sets . They establish generalizations of some fixed point theorems on a class of non-convex sets in a locally bounded topological vector space, and then, obtain Brosowski – Memardus type theorems about random invariant of approximation [9]. Nashine [17] establishes the existence of random fixed point as random best approximations with respect to compact and weakly compact domain .Alsaidy and el at [2] proved coincidence point results for pair of commuting mapping defined on weakly compact separable subset of complete p-normed space. And then , use them to study the random best

approximation in p-normed space with separablity condition[1].

In [fixed point and random best approximations results are proved for random

Banach operator which is defined on separable closed subset of a complete p-normed space. Kirk and Ray [15] have shown that if X is uniformly convex Banach space, A is an unbounded closed convex subset of a and

H :A → A is a Lipschitzian pseudo contractive mapping for which the set G(x, Hx; x) = is bounded for some x A ,

then H has a fixed point inA. Afterward Carbone and Marino [10] employed the structure of some geometric sets in Banach spaces with this property .

Penot [18] and Beg.,and Abbas [7] imposing the condition of asymptotic

contractivity on nonexpansive mappings defined on unbounded closed and convex

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subset of a Banachspace, established some fixed point theorems. Here, we prove some random fixed point theorems for asymptotically nonexpansive random operators defined on an unbounded closed and non-star-shaped subset of a Banach space, where. employ the simple strandom iterative process to obtain the existence of random fixed points of such operators. As a consequence, a stochastic generalization and improvements of the comparable results valid for bounded convex sets in the literature ([12]and [18]) are obtained.

The following classes are needed is the classes of all subsets of ,

is the classes of all bounded ciosed subsets of , is the classes of all weakly compact subsets of , is the weak closure of convex hull of .

2. Preliminaries

Let the pair (Ω, denote to the measurable space with sigma algebra of subsets of Ω and is a metric space.

Definition (2.1):[22]

A mapping : is called measurable (respectively, weaklymeasurable) if, for any closed (respectively,open) subset B of , Definition (2.2): [20]

A mapping is called a measurable selector of a measurable mapping :Ω , if measurable and for each

Definition (2.3): [13]

A mapping (or is called a random operator if for any (respectively G(. , is measurable .

A measurable mapping is called random fixed point of a random operator ,if for every (respectively

A mapping is called demi-closed with respect to y if for each sequence { in such that { converges weakly to x and { converges strongly to y imply that x and .

Definition ( 2.6 ):[7]

Let X be a normed space. A mapping which satisfies the following conditions

i.

ii.

iii.

iv. There is such that For any and .

The mapping G is used to obtain a random fixed point of an uniformly 1-lipschitzain random operators defined on unbounded domain.

Definition ( 2.7 ):[3] ,[19]

Let be a normed space, and be a random operator, then is said to be

i- asymptotically nonexpansive, if there exists a sequence of measurable mappings with for each , such that for arbitrary we have

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‖ , for each ii- uniformly Lipschitzain, if for any , ‖ , where Remark (2.1):[ 19]

Every asymptotically nonexpansive mapping is uniformly Lipschitzain .The random operator is said to be uniformly Lipschitzain iff in definition above.

A random operator is said to be satisfy property ( if for any bounded sequence in with for all implies that

.

Note that, throughout this paper we assume that satisfies the property ( . Definition (2.9):

A set is said to have property ( ) if (i) ,

(ii) , for some and sequence of real numbers ( ) converging to 1 and for each .

Let be a normed space, and be a random operator, is said to satisfy condition ( if for fixed , we have

Definition (2.11): [4]

Let be a normed space, and be a random operator, is said to be asymptotically contractive, if for each there exists with

3. Main results We begin with Theorem ( 3.1):

Let be a separable reflexive Banach space, unbounded closed subset of has property ( ) w.r.t. , be a uniformly Lipschitzain random operatorand be demi-closed for all . If

, then has a random fixed point.

Proof:

Since has property ( ) w.r.t. , then

for some q and a fixed real sequence converging to 1 ( , for all and for all .now, for each defined the random operators by

, for all and all

so, . Since is uniformly Lipschitzain random operator , then

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for all , and all . Therefore each is contractive random operators.

Hence by [8] there is a measurable mapping such that for all …( 3.1.1)

Next, we show that { is bounded sequence for all . Suppose that , for some .

Let and such that

x for all and with ‖x‖ …(3.1.2) From properties of , using (3.1.1) and (3.1.2) we have

‖ ‖

Dividing by and taking ,we have which is contraction. Therefore,{ is bounded sequence for all .

n

Moreover ,

Since { is a bounded sequence in a reflexive Banach space for all ,then for each n , we can define by . Define by .

Since the weakly topology on X is metric topology [Dunford and Schwartz1958,p.434] and the mapping is weakly measurable so Q has measurable selector [Kuratowski and Ryll 1965] .For any fixed in , we may assume that there exists a subsequence { of { converges weakly to

Since converges to zero and is demiclosed for all . Hence has a random fixed point such that .■

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Corollary (3.1) :

Let be a separable reflexive Banach space, unbounded closed and starshaped w.r.t. subset of , be an uniformly

Lipschitzain random operator with and be demiclosed for all . If

, then has a random fixed point . Corollary (3.2) Theorem (3.1)[7]

Let be a nonempty unbounded closed and starshaped subset w.r.t some point in a separable reflexive Banach space and be an asymptotically nonexpansive random operator with and be demiclosed for each . If

, then has a random fixed point . Theorem (3.2):

Let A be a nonempty unbounded closed subset of separable reflexive Banach space X and be an uniformly Lipschitzain with A has property ( ) w.r.t and be demiclosed for all all . If satisfy condition ( , then h has a random fixed point .

Proof :

By similar proof of theorem (3.1) there is measurable mapping such that for all . Now we show that { is bounded sequence for all .

Suppose that ‖ , for some .

Let and such that ‖ for all and with ‖x‖ , consider

Diving by and taking we have which is contraction .therefore is a bounded sequence for each . Hence by similarly proof of Theorem (3.1) there is a measurable mapping such that .■

Corollary (3.3)

Let A be a nonempty unbounded closed and starshaped w.r.t. some in a separable reflexive Banach space X and be an uniformly Lipschitzain with for all in and be demiclosed for all all . If satisfy condition ( , then h has a random fixed point.

Corollary (3.4) theorem (3.3) [7]

Let be a nonempty unbounded closed starshaped subset w.r.t in a separable reflexive Banach space X and be an asymptotically nonexpansine random operator satisfying condition ( with for each . If and is demiclosed for each in , then h has a random fixed point .

Corollary (3.5) corollary (3.4)[7]

Let be a nonempty unbounded closed starshaped subset w.r.t some in a separable reflexive Banach space X and be a nonexpansive asymptotically contractive random operator with for all in Then h has a random fixed point.

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